Английская Википедия:Correlation immunity

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In mathematics, the correlation immunity of a Boolean function is a measure of the degree to which its outputs are uncorrelated with some subset of its inputs. Specifically, a Boolean function is said to be correlation-immune of order m if every subset of m or fewer variables in <math>x_1,x_2,\ldots,x_n</math> is statistically independent of the value of <math>f(x_1,x_2,\ldots,x_n)</math>.

Definition

A function <math>f:\mathbb{F}_2^n\rightarrow\mathbb{F}_2</math> is <math>k</math>-th order correlation immune if for any independent <math>n</math> binary random variables <math>X_0\ldots X_{n-1}</math>, the random variable <math>Z=f(X_0,\ldots,X_{n-1})</math> is independent from any random vector <math>(X_{i_1}\ldots X_{i_k})</math> with <math>0\leq i_1<\ldots<i_k<n</math>.

Results in cryptography

When used in a stream cipher as a combining function for linear feedback shift registers, a Boolean function with low-order correlation-immunity is more susceptible to a correlation attack than a function with correlation immunity of high order.

Siegenthaler showed that the correlation immunity m of a Boolean function of algebraic degree d of n variables satisfies m + d ≤ n; for a given set of input variables, this means that a high algebraic degree will restrict the maximum possible correlation immunity. Furthermore, if the function is balanced then m + d ≤ n − 1.

References

Further reading

1. Cusick, Thomas W. & Stanica, Pantelimon (2009). "Cryptographic Boolean functions and applications". Academic Press. Шаблон:ISBN.

Шаблон:Crypto-stub